confusion in a simple q in corporate finance

I don’t need clearing of air. I do not need this portion of Corp Fin in Level III. I still think it is an erratum if they list the formula with the wrong bracket placement and arrive at 16% Re … which would not come up at all … … and as the OP posted - you would get 14%.

Yea but the basic theory says that the r0 formula doesn’t apply when you already have debt in your structure. Also, the material is clear enough to provide two separate formulas, one for rWACC (which applies to OP’s post) and r0 (which doesn’t apply to OP’s post). Here’s a blurb from the curriculum:

“Because by Proposition I with taxes the value of a company with debt is greater than that of the same company without debt, for the same level of operating income, it must follow that the WACC for the company with debt must be lower than that for the all-equity company. If we continue to define r0 as the cost of capital for an all-equity company , MM show that the cost of equity for the same company with debt is: re = r0 + (r0 - rd)(1 - t)(D/E).”

This formula is used to ascertain a company’s cost of equity when new debt is employed, and taxes are applicable. r0 is the old cost of capital, AKA the cost of equity, because when the company employed no debt, its cost of capital equalled its cost of equity.

If anything, you can reverse engineer r0 based on the inputs of problem 3. Using the rWACC formula, re = 16%. Substituting this value in re = r0 + (r0 - rd)(1 - t)(D/E), you get r0 = 11.11%, the cost of capital if the company employed an all-equity structure. Also, 11.11% is greater than the rWACC = 10%, indicating the advantage of taxes in the new structure.

Anybody who has already registered for L2 this june please send an email to cfa people to clean the air on this issue

THE EQUATION IS NOT WRONG, I’ve seen this question in several forums but I think that Aether could help me with a final remark…

There is no mistake with the brakets neither, the problem is with the set up of the V value:

I’m going to demonstrate it but using the correct expression to V, similar to the one used to derive the unlevered beta V = Dx(1-t) + E

Let’s derive this formula using V = Dx(1-t) + E

Ro = (D/V)xRd(1-t) + (E/V)xRe

Ro - (DxRd(1-t)) / V = (E/V)xRe

Taking LCM and cutting both denominators (V)

VRo - DRd(1-t) = ERe

(VxRo)/E - (DxRd(1-t))/E= Re

Here we go with V = D(1.t)+E, Substituting V with D(1-t)+E

((D(1-t)+E)Ro)/E - (DxRd(1-t))/E = Re

Rewrite it

(D(1-t)/E x Ro) + ERo/E - D/E x Rd(1-t) = Re

(D(1-t)/E x Ro) + ERo/E - D/E x Rd(1-t) = Re

(D(1-t)/E)Ro + Ro - (D/E)Rd(1-t) = Re

Ro + (D(1-t)/E)Ro - (D/E)Rd(1-t) = Re

Taking D/E as common

Ro + (D/E) (Ro - Rd)(1-t) = Re

Rewrite it

Re = Ro + (Ro - Rd)(1-t)D/E

It’s derived in the correct way as Modiggliani Miller proposed first and as is in the book… however I don’t know why they mean that this equation should be used only for an “all-equity company”, ¿what is the mathematic explanation?

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The issue with the problem is weather the capital structure is D / (D+E) = 50% or D / [D(1-t)+E] = 50%

I believe that they gave us the expression for D / (D(1-t)xE) but I have not found the correct answer, so please help me!!!

My thoughts in order to answer the problem assuiming that re = 16% is correct (using reverse engeniering), are below:

In order to have an answer equal to C: re=16, we need that D/E = 1.5 (16% = 10% + (10%-5%)*(1-0.2)*1.5)

So with D/E = 1.5 and t = 0,2 , D/[D*(1-t) +E] will be equal to 0.681818, see how I found it:

If you divide the expresion D/[D*(1-t) +E] by E in both the numerator and denominator you will have (D/E) / [(D/E)*(1-t) +1], and by substituting with D/E = 1.5 and t = 0.2 I arrived to D/ D(1-T)+E = 0.681818

The thing that freaks me out is that there should be a mistake in my calculation or in the way I propose the problem as the solution doesn’t seem to be ok:

WACC = [D / (D(1-t)+E)] x rd x (1-t) + [E / (D(1-t)+E) x re]

10% = 0.681818 x 5% x (1-0.2) + [50% x re]

reordering

re = (10% - [0.681818 x 5% x (1-0.2)] ) / 50%

re = 14.5455%

Would someone help me find a solution without telling me that the formula proposed by MM is wrong and we should change 50 years of papers (and the cfa curriculum too)? I think that the explanation comes from the “all-equity company” expression, but why is it that? Is it because it becomes circular? (you use a given WACC in order to calculate re but when you calculate re taking into account a D/(D+E) > 0 does the WACCchanges?)

tks